TITLE OF INVENTION
AI-POWERED MATHEMATICAL THEOREM VISUALIZATION SYSTEM WITH FORMAL
VERIFICATION AND 3D RENDERING
FIELD OF INVENTION
The present invention relates to the field of computer sci.ence, ~pecifically to artificial
intelligence systems for mathematical theorem proving and visualization. More
particularly, the invention concerns an integrated system that combines automated
theorem proving capabilities with three-dimensional visualization technology to
enhance mathematical reasoning, education, and formal verification processes.
BACKGROUND OF INVENTION
Prior Art and Limitations
Mathematical theorem proving and formal verification have traditionally relied on textbased
systems that require extensive expertise to use effective.y. Existing theorem
provers such as Lean, Coq, and Isabelle provide powerful logical reasoning capabilities
but lack intuitive visualization tools that could make mathematical concepts more
accessible to researchers, students, and practitioners.
Current limitations in the field include:
1. Accessibility Barrier: Traditional theorem provers require extensive training in
formal logic and specialized syntax, limiting their adopti::>n to expert users.
2. Visualization Gap: Existing systems primarily operate in textual domains,
providing limited visual representation of mathematical structures and
geometric relationships.
3. Integration Challenges: Current solutions typically se1=arate theorem proving
from visualization·, requiring users to work with multiple disconnected tools.
4. AI Integration Limitations: Existing theorem provers have limited integration
with modern AI techniques for automated reasoning and proof discovery.
5. Educational Barriers: Mathematical education suffers from lack of interactive,
visual teals that can demonstrate abstract mathematic3l concepts in intuitive
ways.
Need for Innovation
There exists a significant need for an integrated system that combines:
• Automated theorem proving capabilities
• Al-assisted mathematical reasoning
• Interactive 3D visualization of mathematical concepts
• User-friendly interfaces for educational and research applications
• Formal verification tools with visual feedback
SUMMARY OF INVENTION
The present invention provides anAl-powered mathematical theorem visualization
system that addresses the aforementioned limitations through a novel integrated
architecture combining theorem proving, artificial intelligence, and three-dimensional
rendering technologies.
Key Innovations
1. Integrated Architecture: A unified system combining theorem proving engine,
AI reasoning modules, and 3D visualization renderer in a single platform.
2. AI-Enhanced Theorem Proving: Novel algorithms that utilize machine learning
techniques to assist in automated proof discovery and verification.
3. Mathematical Visualization Engine: Advanced 3D rendering system
specifically designed to convert abstract mathematical structures into
interactive visual representations.
4. Functional Programming Integration: Custom functional programming
language optimized for mathematical reasoning and theorem specification.
5. Interactive Educational" Platform: User-friendly interface enabling both novice
and expert users to explore mathematical concepts through visual and
interactive means.
Technical Advantages
• Enhanced Accessibility: Provides visual and interactive interfaces that make
formal mathematics accessible to broader audiences
• Improved Learning: 3D visualization aids in understanding complex
mathematical relationships and structures
• Automated Assistance: AI algorithms provide intelligent suggestions and
automated proof completion
• Formal Verification: Maintains mathematical rigor while providing intuitive user
interfaces
• Scalability: Modular architecture supports extension to various mathematical
domains
DETAILED DESCRIPTION OF INVENTION
System Architecture
The Theorem AI system comprises several interconnected components working in
harmony to provide comprehensive mathematical reaso'ning and visualization
capabilities:
1. Core Theorem Proving Engine
The central theorem proving engine processes mathematical statements written in the
Theorem AI functional programming language. This engine implements:
• Type System: Advanced dependent type system ensuring mathematical
correctness
• Proof Checker: Rigorous verification of mathematical proofs and derivations
• Inference Engine: Automated reasoning capabilities for proof discovery
• Pattern Matching: Sophisticated pattern matching for mathematical
expressions
2. AI Reasoning Module
The AI reasoning module enhances the theorem proving capabilities through:
• Machine Learning Integration: Neural networks trained on mathematical
corpora for proof suggestion
• Heuristic Search: Intelligent search algorithms for proof space exploration
• Automated Lemma Discovery: At-assisted identification of useful intermediate
results
• Context-Aware Reasoning: Adaptive reasoning based on mathematical domain
and user expertise
3. 3D Visualization Engine
The visualization engine converts mathematical structures into interactive 3D
representations:
• Geometric Interpretation: Algorithms for translating logical structures into
geometric forms
• Real-time Rendering: High-performance 3D graphics rendering with real-time
interaction
• Mathematical Accuracy: Precise representation maintaining mathematical
relationships
• Interactive Manipulation: User interfaces for exploring mathematical objects in
3D space
4. Integration Layer
The integration layer coordinates between all system components:
• Data Flow Management: Efficient communication between theorem prover and
visualizer
• State Synchronization: Consistent state management across different system
modules
o User Interface Coordination: Unified user experience across text-based and
visual interfaces
• Extension Framework: Plugin architecture for adding new mathematical
domains
Implementation Details
Functional Programming Language
The system includes a custom functional programming language specifically designed
for mathematical expression:
--Example: Factorial function definition
def factorial (n: Nat) : Nat:=
match n with
I o => 1
1 n + 1 => (n + 1) *factorial n
Key language features:
o Dependent Types: Types that depend on values, enabling precise mathematical
specifications
o Pattern Matching: Intuitive syntax for defining functions over mathematical
structures
• Proot'Terms: First-class support for mathematical proofs as programming
constructs
• Type Inference: Automatic deduction of types reducing user annotation burden
AI Integration Mechanisms
The system integrates artificial intelligence through several mechanisms:
1. Proof Suggestion Network: Neural network trained to suggest next steps in
proof development
2. Theorem Classification: Machine learning models for categorizing
mathematical statements
3. Automated Proof Search: Al-guided search through proof spaces using
reinforcement learning
4. Natural Language Processing: Conversion between natural language
mathematical statements and formal representations
Visualization Algorithms
The 3D visualization system employs novel algorithms for mathematical visualization:
1. Structural Mapping: Algorithms that map logical structures to geometric
representations
2. Dynamic Rendering: Real-time updates of visualizations as mathematical
structures evolve
3. Interactive Exploration: User interface elements allowing manipulation of
mathematical objects
4. Multi-scale Visualization: Techniques for visualizing mathematical structures
at different levels of detail
System Components
Build System and Development Environment
• CMake Integration: Cross-platform build system supporting Windows, macOS,
and Linux
• Modular Architecture: Separated core libraries enabling independent
development and testing
• Standard Library: Comprehensive mathematical library covering various
domains
• Testing Framework: Comprehensive test suite ensuring system reliability
User Interfaces
• Interactive Mode: Command-line interface for direct interaction with theorem
prover
• Batch Processing: Capability to process multiple files and complex proofs
• Visual Interface: Graphical user interface integrating 3D visualization with
theorem proving
• Educational Mode: Simplified interface designed for learning and teaching
mathematics
INDUSTRIAL APPLICABILITY
The present invention has significant industrial applicability across multiple sectors:
'
Education Sector
• Mathematical education at university and research levels
• Interactive textbooks and learning materials
• Online mathematical course platforms
• Teacher training and educational tool development
Research and Development
• Mathematical research acceleration through AI assistance
• Formal verification in software and hardware development
• Scientific computing and mathematical modeling
• Algorithm development and optimization
Technology Industry
• Software verification and validation
• Cryptographic protocol verification
• AI system development and testing
• Mathematical software development tools
Commercial Applications
• Educational software licensing
• Professional mathematical tool development
• Consulting services for mathematical problem solving
• Training and certification programs
' ' The invention addresses substantial market needs in mathe'm atical education,
research tools, and formal verification systems, providing significant economic and
educational benefits.

1/we CLAIMS that,
Claim 1 (Overall Claim)
Computer-executed artificial intelligence mathematical theorem proving and threedimensional
visualization system of:
a) A computer program for theorem proving in design to manage mathematical
statements and build proofs using a functional programming language with
dependent types;
b) An artificial intelligence reasoning module configured to provide automated proof
assistance using machine learning algorithms trained on mathematical corpora;
. c) Three-dimensional visualization engine that is coded to translate mathematical
structures into 3D geometric, interactive figures;
d) An integration layer that is established to handle data transmission among
theorem proving engine, AI reasoning module, and visualization engine;
e) A user interface configured to provide unified access to theorem proving and
visualization capabilities;
in which the graphical representations of mathematical theorems are generated
automatically while attaining formal verification of mathematical correctness.
Claim 2 (Dependent Claim)
System as recited in claim 1, wherein the functional programming language includes:
• Dependent type system of mathematical specification
• Pattern matching functions in function definition
Intrinsic proof term support
• Automatic type inference mechanisms
Dependent Claim 3 (Claim
The system of claim 1, such that the AI reasoning module includes:
• Propose step suggestion models by using neural networks
• Proof searching optimization algorithms by reinforcement learning
• . Heuristic evaluation is done under proof state testing
• Context-aware reasoning attuned to mathematical domains
Claim 4 (Dependent Claim)
System by claim 1, wherein said 3D visualization engine includes:
shapes
• Real-time rendering capabilities with interactive manipulation
Scale-level visualization techniques operating at different scales of
mathematical abstraction
• Mathematical correctness of visualizations being preserved
Claim 5 (Method Claim)
Computer program product to visualize mathematical theorems under AI influence
. comprising:
a) Input of mathematical statements in functional programming language notation; b)
Procesing the statements by theorem proving engine with formal verification; c)
Utilizing AI reasoning algorithms as support in proof building; d) Creating threedimensional
geometric models of mathematical structures; e) Producing interactive
visualizations with mathematical integrity; f) Offering user interface for mathematical
concept manipulation and exploration;
Which brings automated theorem proving and visual mathematical representation
together.
Claim 6 (System Architecture Claim)
Modular software design of mathematical reasoning and visualization in terms of:
• Standard core theorem proving libraries with type checking and inference
support
•_AI reasoning modules with machine learning integration_
• Subsistema de renderizado de graficos 3D con
• Plugin-based architecture of mathematical landscape extension
•
• Cross-platform build system supporting multiple operating systems
Claim 7 (Educational Application Claim)
A learning system by claim 1, especially developed for mathematical studies,
comprising:
• Intuitive user interfaces for non-expert end-users
Interactive tutorials with mathematical visual demonstrations
• Increasing degree of abstraction in mathematical concept description
• Automatically verified proof assessment tools
Shared mathematical exploration collaborative features
Claim 8 (AI Enhancement Claim)
8
The AI reasoning module of claim 3, further comprising:
Transfer learning characteristics to suit new mathematical fields
• Algorithms of automated lemm
• Proof complexity estimation mechanisms
Intelligent error identification and correction advice·
Claim 9 (Visualization Enhancement Claim)
It
The 30 visualization engine by claim 4, further including:
• Facilitation of mathematical experimentation in immersive virtual reality
• Real-world demonstration of mathematical augmented reality functions
• Collaborative visualization features for multiple user interaction
• Functions of animation in mathematical transformation presentation
Claim 10 (Integration Claim)
A data synchronisation system to synchronise proof-theoretic and visualisation
aspects, comprising:
• Synchronous State Management Between Logical and Visual Models
•_ Two-way communication protocols among system parts_
• Checking protocols that ensure mathematical soundness
Real-time performance optimization of interactive visualization